Circuit complexity is closely related to operator size which measures the number of d.o.f. that an operator can change. In a quantum circuit, operator size is the derivative of the circuit complexity w.r.t. the circuit time [1]}. Inspired by this relation, we adopt the average operator size as the distance function to define complexity. To test whether it is an appropriate definition, a basic criterion is that the complexity geometry should be defined on negatively curved manifolds [2]}, [1]}, [4]}. However, we find that the typical sectional curvatures are positive, even if we replace the complexity metric by an arbitrary function of operator size.
